Differentiate `sin(xy) + (x)/(y) = x^(2) - y`

Explanation

To differentiate the equation \( \sin(xy) + \frac{x}{y} = x^2 - y \) with respect to \( x \), we will follow these steps:

### Step 1: Differentiate both sides
We start by differentiating each term in the equation with respect to \( x \):

\[ \frac{d}{dx}(\sin(xy)) + \frac{d}{dx}\left(\frac{x}{y}\right) = \frac{d}{dx}(x^2) - \frac{d}{dx}(y) \]

### Step 2: Apply the chain rule and product rule
For the first term \( \sin(xy) \), we use the chain rule:

\[ \frac{d}{dx}(\sin(xy)) = \cos(xy) \cdot \frac{d}{dx}(xy) = \cos(xy) \cdot (y + x\frac{dy}{dx}) \]

For the second term \( \frac{x}{y} \), we apply the quotient rule:

\[ \frac{d}{dx}\left(\frac{x}{y}\right) = \frac{y \cdot \frac{d}{dx}(x) - x \cdot \frac{d}{dx}(y)}{y^2} = \frac{y \cdot 1 - x \cdot \frac{dy}{dx}}{y^2} = \frac{y - x\frac{dy}{dx}}{y^2} \]

For the right side:

\[ \frac{d}{dx}(x^2) = 2x \quad \text{and} \quad \frac{d}{dx}(y) = \frac{dy}{dx} \]

### Step 3: Combine the results
Now we can combine all these results into our differentiated equation:

\[ \cos(xy) \cdot (y + x\frac{dy}{dx}) + \frac{y - x\frac{dy}{dx}}{y^2} = 2x - \frac{dy}{dx} \]

### Step 4: Rearrange the equation
Next, we will rearrange the equation to isolate \( \frac{dy}{dx} \):

\[ \cos(xy) \cdot y + \cos(xy) \cdot x\frac{dy}{dx} + \frac{y - x\frac{dy}{dx}}{y^2} = 2x - \frac{dy}{dx} \]

### Step 5: Collect all \( \frac{dy}{dx} \) terms
Now, we will collect all terms involving \( \frac{dy}{dx} \):

\[ \cos(xy) \cdot x\frac{dy}{dx} + \frac{y}{y^2} - \frac{x}{y^2}\frac{dy}{dx} = 2x - \cos(xy) \cdot y + \frac{dy}{dx} \]

### Step 6: Factor out \( \frac{dy}{dx} \)
This gives us:

\[ \left(\cos(xy) \cdot x - \frac{x}{y^2} + 1\right)\frac{dy}{dx} = 2x - \cos(xy) \cdot y \]

### Step 7: Solve for \( \frac{dy}{dx} \)
Finally, we can solve for \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \frac{2x - \cos(xy) \cdot y}{\cos(xy) \cdot x - \frac{x}{y^2} + 1} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = \frac{2x - \cos(xy) \cdot y}{\cos(xy) \cdot x - \frac{x}{y^2} + 1} \]

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